Unveiling EIlenberg-type correspondences: Birkhoff's theorem for (finite) algebras + duality

The purpose of the present paper is to show that: Eilenberg{type correspondences = Birkho's theorem for (nite) algebras + duality. We consider algebras for a monad T on a category D and we study (pseudo)varieties of T{ algebras. Pseudovarieties of algebras are also known in the literature as varieties of nite algebras. Two well{known theorems that characterize varieties and pseudovarieties of algebras play an important role here: Birkho's theorem and Birkho's theorem for nite algebras, the latter also known as Reiterman's theorem. We prove, under mild assumptions, a categorical version of Birkho's theorem for (nite) algebras to establish a one{to{one correspondence between (pseudo)varieties of T{algebras and (pseudo)equational T{theories. Now, if C is a category that is dual to D and B is the comonad on C that is the dual of T, we get a one{to{one correspondence between (pseudo)equational T{theories and their dual, (pseudo)coequational B{theories. Particular instances of (pseudo)coequational B-theories have been already studied in language theory under the name of \varieties of languages" to establish Eilenberg{type correspondences. All in all, we get a one{to{one correspondence between (pseudo)varieties of T{algebras and (pseudo)coequational B{theories, which will be shown to be exactly the nature of Eilenberg{type correspondences.